WMA13 Pure Mathematics 3 (P3): Oct/Nov 2025 Paper 01

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The functions f and g are defined by

(a)State the range of f[1]

(b)Find g^–1[3]

(c)Find fg(0)[2]

(d)Find the exact value of a for which

[3]

y  ( (x)

Figure 1 shows a sketch of part of the curve with equation y = f (x) where

(a)Show that f (x) = 0 has a root α in the interval [1, 2][2]

(b)Show that the equation f (x) = 0 can be written in the form

[2]

Using the iteration formula

find, to 4 decimal places,

(c)(i)the value of x₃

(ii)the value of α[3]

The share price, £V, of a company is being monitored.

A graph is drawn of log₁₀ V against t, where t is the number of years after monitoring began.

The graph is a straight line passing through the points (0, 2) and (5, 2.25)

Using this information,

(a)find an equation for the line in the form

where m and c are constants.[2]

(b)Write the answer to part (a) in the form

where a and b are constants to be found.

Give the exact value of a and the value of b to 3 significant figures.[3]

When t = T, the rate of increase in the share price of the company was £50 per year.

(c)Find the value of T, giving your answer to the nearest integer.

[4]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

(a)f (x) = 3 sin 2x – 3 cos 2x

Express f (x) in the form R sin (2x – α) , where R and α are constants,

Give the exact value of R and the exact value of α.[3]

(b)g(x)  x > 0

Using the answer to part (a),

(i)write down the exact minimum value of g(x) ,

(ii)find the smallest value of x for which this minimum value occurs.

You must make your method clear.[3]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The number of squirrels in a forest is being studied.

The number of squirrels, N, in the forest, t years after the start of the study, is modelled by the equation

(a)Find the number of squirrels in the forest at the start of the study.[1]

(b)Find dNdt[2]

The number of squirrels in the forest is at a maximum when t = T

Using the answer to part (b),

(c)show that e^0.2T = A , where A is a constant to be found.[2]

(d)Hence find the maximum number of squirrels in the forest.

Show your working and give your answer to the nearest whole number.[3]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

(i)Given y = ln (2x² + 5) find dydx[2]

(ii)A curve C has equation y = f (x) where

where k is a positive integer, k > 1

Given that

find the greatest possible value of k[5]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A curve C has equation

where k is a constant.

Given that C has two distinct turning points, find the range of possible values of k.[7]

Figure 2 shows a sketch of the graph C₁ with equation

and a sketch of the graph C₂ with equation

where a and b are constants.

Graphs C₁ and C₂ intersect at point P, point Q and point R, as shown in Figure 2.

Given that P has coordinates (–2, 25)

(a)show that

[2]

Given also that R has coordinates (2, 9)

(b)find the value of a and the value of b[3]

Using the answer to part (b),

(c)state the coordinates of the vertex of C₂[2]

(d)Find, using algebra, the coordinates of Q. Show each stage of your working.

[6]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

(a)Express

in terms of sin 2θ and cos 2θ only.[3]

(b)Hence show that the equation

can be written in the form

[3]

(c)Hence solve, for 0 < x < 90° , the equation

giving your answers to one decimal place.[3]